Does the Series Sum of e^n/(1+e^(2n)) Converge?

[karush]

$\tiny{10.3.35}\\$
$\textsf{Does $S_n$ converge or diverge?}\\$
\begin{array}{lll}
&S_n&=&\displaystyle
\sum_{n=1}^{\infty}\frac{e^{n}}{1+e^{2n}}\\
\end{array}
$\textsf{presume we could use Imtegral test}$

 

[MarkFL]

Re: 10.3.35 converge or diverg

Let's try the ratio test:

$$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim_{n\to\infty}\left|\frac{\dfrac{e^{n+1}}{e^{2(n+1)}+1}}{\dfrac{e^{n}}{e^{2n}+1}}\right|=e\lim_{n\to\infty}\left(\frac{e^{2n}+1}{e^{2(n+1)}+1}\right)=e\lim_{n\to\infty}\left(\frac{2e^{2n}}{2e^2e^{2n}}\right)=\frac{1}{e}<1$$

Therefore, the series is convergent. :D